Corruption as an Alternative to Limit Pricing. Raluca Elena Buia Università Ca Foscari di Venezia

Transcription

1 Working Papers Department of Economics Ca Foscari University of Venice No. 02/WP/2011 ISSN Corruption as an Alternative to Limit Pricing Raluca Elena Buia Università Ca Foscari di Venezia First Draft: 21/10/2010 Abstract We explore to what extent bribery can be an alternative way of fighting rivals entry on the market when there is uncertainty about the degree of corruption in the public sector. For high levels of corruption, covert fight through bribery is the optimal choice of an incumbent. For low degree of corruption, instead, the incumbent prefers to act strategically but overtly by playing a limit pricing game. Keywords Corruption, Bribery, Production licence, Moral cost, Covert/overt fight JEL Codes D21, D73, H40 Address for correspondence: Raluca Elena Buia Department of Economics Ca Foscari University of Venice Cannaregio 873, Fondamenta S.Giobbe Venezia - Italy Phone: (++39) Fax: (++39) elenabui@unive.it The Working Paper Series is availble only on line ( For editorial correspondence, please contact: wp.dse@unive.it Department of Economics Ca Foscari University of Venice Cannaregio 873, Fondamenta San Giobbe Venice Italy Fax:

2 This Working Paper is published under the auspices of the Department of Economics of the Ca Foscari University of Venice. Opinions expressed herein are those of the authors and not those of the Department. The Working Paper series is designed to divulge preliminary or incomplete work, circulated to favour discussion and comments. Citation of this paper should consider its provisional character. Testo WP Testo WP 2

3 Corruption as an Alternative to Limit Pricing October 21, Introduction The literature on corruption has been receiving much attention in the last two decades, after the end of the Cold War. However, the research topics are far from being exhausted. Corruption is an extremely complex phenomenon with many aspects and implications. One controversial issue in the field refers to the interdependencies between corruption and competition. The starting point in some of the existing studies is that the existence of corruption is strongly related to the existence of rents, for example Ades and Di Tella (1996, 1999), Tanzi (1998), Arnone and Iliopulis (2005). Certainly, if we speak about goods allocation, rents appear or are larger where the competition is less strong. Consequently, the conclusion of several studies is that the stronger the competition, the lower the registered levels of corruption. This idea finds support in the empirical evidence, for example Ades and Di Tella (1996, 1999) or Gurgur and Shah (2005). However, the relationship is not as clear as it seems. Bliss and Di Tella (1997) try to explain the relationship between the competition level and corruption in a model in which competition is endogenous. They find that the existence of rents is not a necessary premise for corruption to exist and operate. Corruption can appear also in markets characterized by perfect competition. This happens because corruption is able to generate rents by inducing less-efficient firms to exit the market. On the same ground of the interactions between corruption and competition, the purpose of this paper is to address the question: can corruption become a barrier to entry of new competitors? And if yes, to what extent an incumbent prefers a covert bribery behavior to an overt strategically fighting entry. There is no doubt that corruption is strongly related to decision power. In our paper a public (corruptible) official is delegated the task and hence the power to issue or deny the production licenses for the potential entrants on the market. We assume that any public employee is corruptible if he is offered a sufficiently large amount of money. However, the moral cost that the public employee has when behaving dishonestly is private information and this dictates the amount of bribe necessary to persuade him to favor the graft-payers. Departing from the models in the existing literature, in which the corrupt employee is asking a 1

4 bribe to the potential interested party (Bliss and Di Tella 1997), in our paper the interested himself proposes a bribe but he does not have the certainty of the graft being accepted. The main results show that in economies where the public sector is characterized by low corruption an incumbent prefers to act strategically but overtly by playing a limit pricing game. Hence, at low levels, corruption does not affect competition and market outcomes. In an economy with high levels of corruption, instead, the incumbent finds the bribing alternative to be the most attractive, especially when the market is large and/or the cost of installing production capacity is low. The paper is organized as follows: in Section 2 we introduce the model with the main assumptions. Sections 3 and 4 contain the descriptions with the main outcomes of the covert and overt behavior respectively. Section 5 analyzes the optimal choices of a profit-maximizing incumbent. Section 6 briefly explores the social implications of a bribery behavior. In section 7 we present two extensions of our model. Finally, Section 8 states the conclusions of the paper. 2 The Model We consider a market characterized by the inverse demand function: p = A q. Initially, there is only one firm (hereafter the incumbent) operating in the market. Entry of new firms is conditional on their obtaining a production licence certifying that all legal requirements (environmental constraints, quality or safety requirements etc.) are fulfilled. The entrant has to pay a fixed cost, F, representing a lump-sum tax, in exchange for the production licence 1. A public official is delegated to issue the production licenses. In order to produce one unit of the good any firm needs one unit of capacity and one unit of labor (firms face capacity constraints). One unit of capacity costs c 0 while we assume, for convenience, that labor cost is 0. We assume that the incumbent can invest in production capacity in a preliminary stage. Investments in production capacity are sunk costs and make credible the incumbent s commitment to given levels of production. The incumbent s cost function is: C i (q i ) = c 0 max {0, q i k} + I k where I k is the investment in capacity k (sunk cost). In words, any quantity of the good q has a marginal cost of 0 if it is inferior or equal to the existent production capacity (q k) while producing a quantity q > k implies a marginal cost of c 0 for every unit of the good additional to the production capacity k. The entrant s cost function is: C(q) = F + c 0 q e. The public employee may be corruptible and, in exchange for a bribe, he can favor the graft-payers. The utility that the public official gets from receiving grafts is: U = B λ 1 This fee is paid when the production licence is issued. Note that it is not a sunk cost but a fixed cost. 2

5 where B is the amount of the bribe that he receives while λ is the moral cost of behaving dishonestly. The parameter λ is private information but it is common knowledge that λ U[0, Λ]. Note that the higher is Λ, the less corruptible is the employee; so, Λ is inversely related to corruptibility. In what follows we assume that there is one candidate (potential) entrant which satisfies all legal requirements for entering the market. The assumption that the entrant satisfies the legal requirements is not restrictive but it is part of the analysis, because we study here the influence that corruptibility of the public employee can have on the entry of qualified firms 2. In what follows we restrict attention, for convenience, to the analysis of a situation in which the only player who can offer grafts is the incumbent. In a later extension we prove that the results are similar, with little variations, also for the case in which both the incumbent and the entrant can offer bribes (when they compete in bribes). The timing of the game is the following: At date 1 the incumbent can choose between acting: i) overtly by playing a Dixit (1980) limit pricing game (commit to given levels of production by installing production capacity k); or ii) covertly by offering a graft in order to persuade the public official not to issue the production licence for the potential entrant. Once he decides for an action, he cannot switch to the other one in case of unsatisfactory result. Hence, in case the bribery game fails to deter entry, the incumbent cannot anymore install production capacities before the market game starts (cannot play anymore overt fighting through commitment). The consequence of this assumption will be clearer after we solve the game. At date 2, based on the incumbent s first move, the overt or covert subgame is played. We describe in detail the covert and the overt subgames and their outcomes, separately, in the following sections. Throughout the paper we assume, for convenience, that c 0 < 1 2A. Such an assumption is useful when studying the Dixit (1980) model called hereafter the overt fight. 3 Covert fight In this subgame the incumbent offers a bribe to the public official in order to persuade him not to issue the production licence for a candidate entrant. The game is sequential: in the first stage the incumbent offers a bribe to the public official. In the second stage the public employee decides whether to accept or reject the graft. If the employee accepts, he denies the production licence 3, entry does not occur and the incumbent maintains a monopolistic position on 2 In another paper we study the choice of qualifying or not when the public employee is corruptible. 3 He can find reasons to (postpone) block the entry. 3

6 the market. If, instead, the public employee rejects the bribe offer, he issues the production licence and Cournot competition is played. The choice between accepting or rejecting the bribe depends on the official s utility function and it is immediate that the public official acts dishonestly if: that is, when: U(accept) U(reject); B λ 0 λ B This means that the public employee chooses to act dishonestly when his monetary gain from behaving so is greater than the moral cost of adopting such a behavior. When the bribe is paid, the incumbent remains the only producer in the market and hence the monopoly outcome occurs. When the public official rejects the graft, entry occurs and each of the two firms gets the Cournot profit. Note that in such a case competitors are on equal positions, namely both have the same cost function, since the incumbent cannot anymore install production capacity before market game starts. Overall, the incumbent expects a level of profits of: EΠ i = P r(λ B) (Π m B) + P r(λ > B) Π c where Π m stands for monopoly profits while Π c represents the Cournot profits when none of the firms have previously installed capacity (k = 0). Given our assumptions this becomes: EΠ i = B ( Λ (Πm B) + 1 B ) Π c Λ and from the first order condition we get the optimal bribe: Substituting for Π m and Π c we get: B = 1 2 (Πm Π c ) B = 5 72 (A c 0) 2 Note that the optimal bribe is increasing with A (the dimension of the market) and decreasing in c 0. This is due to the fact that the incumbent s gain from deterring entry is increasing with A but decreasing in c 0. We can easily observe that both Π m and Π c increase with the difference (A c 0 ) but Π m increases faster than Π c does. This makes it worth to the incumbent to pay a higher bribe which guarantees him a higher probability of deterring entry and maintaining a monopoly position. Based on the relationship between Λ e 5 72 (A c 0) 2, it is useful to distinguish two situations: 4

7 Λ < 5 72 (A c 0) 2. In this case, offering a bribe of amount 5 72 (A c 0) 2 is dominated by offering a graft of Λ. We remind that Λ represents the upper bound of the moral cost of behaving dishonestly (the maximum possible moral cost of behaving dishonestly). This means that any public official has at most a moral cost of Λ. So, offering a graft of Λ compensates for sure any type of public official for his moral costs of behaving dishonestly. Consequently, if a graft Λ is offered, it is accepted with certainty and hence, the entrant s application for production licence is rejected. This means that paying Λ < 5 72 (A c 0) 2 makes the incumbent better off than 5 paying a graft of 72 (A c 0) 2. Λ 5 72 (A c 0) 2. In this case the optimal bribe to be offered is B = 5 72 (A c 0) 2. We study both cases in more detail when we do the comparative analysis in Section 5. Given the above considerations, in order to have a complete and correct expression for the optimal bribe we redefine it as below: { B = min Λ, 5 } 72 (A c 0) 2 (1) Then, given a dishonest behavior of the incumbent, the probability that entry is blocked by a corrupt public official is: { } 5 P r(deny p.l.) = P r(b λ) = min 72Λ (A c 0) 2, 1 4 Overt fight In this subgame, the incumbent optimally chooses the production capacity to install. Practically, such a decision is equivalent to choosing the best action among the following possibilities: fighting the entrant (by playing limit pricing) or strategically accomodating entry (by playing Stackelberg or Cournot whichever is optimal for the incumbent). In what follows, we compute the optimal production capacity k in various scenarios, using the Dixit (1980) model 4. Intermediary results, namely the outcomes of the different situations (monopoly, Cournot without capacity commitments (k i = 0), Cournot when the incumbent s capacity is k = and Stackelberg), necessary in order to compute k are reproduced in the Appendix. Based on the entrant s level of profits, we distinguish three main situations: 1. Blockaded monopoly: In this case the entrant has negative profits in the common Cournot competition. The limit quantity is lower than q c i and, consequently, the incumbent can act as a monopolist while still deterring entry. 4 We follow the approach of Tirole (1994) and Church and Ware (2001). 5

8 Producing the monopoly quantity and hence installing the monopoly capacity is the incumbent s optimal action in this case. k = q S i where qi S is the optimal quantity of a leader under Stackelberg competition. This situation can occur for high levels of the entry fee, F, namely when: F 1 9 (A c 0) 2 2. Strategic deterrence/accomodation: In such a case the entrant s profits are positive in a Cournot game in which players are on equal positions (players have the same marginal cost - none of the players have positive production capacity before the game starts) but negative in the equilibrium of a game in which the incumbent has an infinite production capacity 5. This means that the limit quantity is comprised between qi c and qck i (see Appendix), where qi ck is the optimal supply of the incumbent in a Cournot competition in which the incumbent has infinite production capacity 6. Such a situation happens when: 1 9 (A 2c 0) 2 F < 1 9 (A c 0) 2 that is, when the entry fee is significantly high, though not as large as in the previous situation. In such a case the incumbent could profitably play limit pricing but, since his final purpose is (not entry deterrence but) profit maximization he chooses that action which maximizes his payoff. Based on the incumbent s optimal action we distinguish the following situations: Blockaded monopoly: This situation, which happens for levels of the entry fee that simultaneously satisfy: { 1 max 9 (A 2c 0) 2, 1 } 16 (A c 0) 2 F < 1 9 (A c 0) 2 is characterized by the fact that the limit quantity is lower than the monopoly output. Consequently, it becomes optimal for the incumbent to install the monopoly capacity and produce the monopoly output while successfully deterring entry. Limit pricing: The limit quantity becomes optimal (k = q l ) in situations in which the entrant has to pay a significantly positive entry fee but still F is not very large: { max (A 2c 0) 2, (3 2 } 2) (A c 0 ) 2 32 F < 1 16 (A c 0) 2 5 The Cournot game when the incumbent s production capacity is zero, is different from the Cournot game in which the incumbent capacity is infinity because in the two games the incumbent has different cost functions (see the description of the model on page 2). 6 This is equivalent to the incumbent having a marginal cost of 0 while the marginal cost of the entrant remains c 0. 7 This is 1 9 (A 2c 0) 2 for A > 2, 28c 0 and (A c 32 0 ) 2 otherwise. 6

9 In such a situation, the incumbent s commitment to the monopoly output does not keep the entrant out of the market. The level of the entry fee, although significant, still allows the entrant to obtain positive profits if the incumbent produces his monopoly quantity. Note that such a situation can occur only if c A 8. Stackelberg optimality: This case arises only for significantly high capacity costs 9 c 0, and the levels of entry fee that support Stackelberg are: 1 9 (A 2c 0) 2 F < (3 2 2) (A c 0 ) 2 32 Note that Stackelberg optimality is induced by a situation in which the entry fee is positive but low. This is due to the fact that the level of the entry fee has a direct influence on the limit quantity. The limit quantity, q l is negatively related to F (see Appendix for the expression of q l ) and hence a low entry fee induces a large limit quantity, q l and, consequently, a low price. Then, for high levels of the capacity cost, c 0, the additional gains from deterring entry may be offset by the higher sunk costs for installing capacity q l. Consequently, when costs c 0 are large while the entry fee F, is rather small, it becomes optimal for the incumbent to strategically accomodate entry by playing Stackelberg instead of deterring entry at all costs. 3. Optimal accomodation: This is the situation in which the profits of the potential entrant are positive in the worst possible post-entry equilibrium (when the incumbent s installed capacity is unlimited, k = ). Optimal accomodation is supported by very low levels of the entry fee: F < 1 9 (A 2c 0) 2 In such a situation, entry deterrence (limit pricing) is never an optimal choice for the incumbent who chooses his optimal action among the possible ways of accomodating entry. The reason is, again, that the low entry fee (determining a high limit quantity 10 and correspondingly a low limit price) make the limit quantity an alternative with extremely high opportunity cost even when c 0, the capacity installing cost, is low. The additional gains from deterring entry are not large enough to compensate for the decrease in price that accompanies large levels of q l. 8 This condition assures that 1 9 (A 2c 0) 2 < 1 16 (A c 0) 2. When c 0 < 1 A, instead, all 5 values of the entry fee considered promote q m as optimal strategy. 9 Only for those values of c 0 for which 1 9 (A 2c 0) 2 < (3 2 2) (A c 32 0 ) 2, namely for c 0 > A( A) In this situation we have that q l > q ck. We remind that q ck is the optimal quantity to be produced by a firm in a Cournot game in which the firm has an infinite production capacity (or a marginal production cost of 0) while the rival has no installed capacity (or a marginal cost of c 0 ). 7

10 This situation also includes the special case in which there is no entry fee. Depending on the incumbent s optimal move we identify two different situations: Stackelberg optimality: This case, which can occur only for large capacity costs: c 0 > 1 5A, is the situation in which the incumbent s profits are maximized by playing Stackelberg and is practically characterized by q1 c < q1 S < q1 ck. Cournot dominance: Finally, this is the case in which playing Cournot with capacity commitments, that is, installing capacity k = q1 ck and producing q1 ck is the best choice of the incumbent. This is optimal only when c 0 is low: c 0 < 1 5 A, and hence qs 1 > q ck 5 Optimal choice and main outcomes We identify now the situations in which overt fight is optimal for the incumbent and those that, on the contrary, support covert fight as the best incumbent s strategy. In addition, we establish, on one hand, in which circumstances entry occurs and, on the other hand, to what extent the degree of corruption negatively influences competition. In what follows, we separately analyze the two cases identified in section 3, namely the situation when the perceived degree of corruption is medium-low and the situation when corruption is high, respectively. 5.1 Medium-low corruption, Λ 5 72 (A c 0) 2 In order to simplify the analysis we group together all the cases in which the incumbent behaves in the same way, no matter which are the causes that determine such a behavior. Blockaded monopoly versus bribe: As we discussed above, we have blockaded monopoly when the entry fee is significantly high, inducing a very low level of the limit quantity q1 l < q c or q1 l < q m : { 1 F max 9 (A 2c 0) 2, 1 } 16 (A c 0) 2 Then, the incumbent s best choice is to install the monopoly capacity (k = q m ) and to produce at the monopoly level. Such a behavior deters entry while maximizing the incumbent s payoffs. This is the simplest possible situation because the incumbent cannot be better off by behaving dishonestly. Hence, we can assert that for high levels of the entry fee the incumbent s optimal strategy is to fight overtly. Proposition 1. For very high levels of the entry fee a covert fight is never optimal for an incumbent. 1. 8

11 Limit quantity versus bribe: When limit quantity is the optimal legal action, { 1 max 9 (A 2c 0) 2, (3 2 } 2) (A c 0 ) 2 F < (A c 0) 2 and c 0 > 1 5 A. Computing the difference between the expected profits from a covert fight and the profits from an overt fight we get: EΠ i Π l i = 4F 2 F (A c 0 ) Λ (A c 0) (A c 0) 2 It is easy to observe that the difference EΠ i Π l i is negatively related to the level of the upper-bound, Λ: the higher is Λ, the lower is the probability that the public employee accepts the bribe and blocks entry and hence the lower is the relative profitability of a dishonest behavior (recall that Λ is the maximum moral cost to the public official from behaving dishonestly). Indeed, we find out that: EΠ i Π l i 0 only for: 25(A c 0 ) 4 Λ 5184 ( 4F + 2 ) F (A c 0 ) 19 (A c 0) 2 The right hand side of the above expression is a threshold and we denote it by Λ l. Note that this is strictly decreasing with the value of the entry fee F : the higher is F, the lower is the range of the values Λ that induce covert behavior for the incumbent. For a given value of the entry fee, the incumbent finds it optimal to behave dishonestly only if such values of the entry fee are accompanied by sufficiently low levels of Λ (that is, high probability of corruption). The intuition behind this is the following: a high entry fee determines a low limit quantity q l, and this implies that an overt fight is not very costly. Hence, under the circumstances, the incumbent prefers a covert behavior to overtly playing q l only if the probability of success of the bribing strategy is high (Λ is low). If the value of Λ is high the incumbent finds it optimal to play overtly. Before any further comments, we remind that we are here in a situation in which the level of corruption is not high (we assumed in this case that Λ 5 72 (A c 0) 2 ) so we check if in such a case the incumbent ever acts covertly. By comparing the domain of Λ with the threshold Λ l we find out that Λ l 5 72 (A c 0) 2 for those values of F that satisfy: 6 10 F (A c 0 ) 0, 118(A c 0 ) 24 while the threshold is always lower otherwise. We can thus affirm that the incumbent finds it preferable to play dishonestly only if the entry fee F is low and, at the same time, the moral cost of the public official is not too high: 5 72 (A c 0) 2 Λ 25(A c 0 ) ( 4F + 2 ) F (A c 0 ) 19 (A c 0) 2 9

12 We remark that, in this case the perception of relatively high corruption may be beneficial in the sense that it switches the behavior of the incumbent from strategically playing limit pricing, with zero probability of entry, to bribery, which guarantees a positive probability of entry. Just in order to complete the analysis we add that for a significantly high entry fee, F (A c 0 ) 0.118(A c 0 ), it is always verified that: 5 max 72 (A c 0) 2 25(A c 0 ) 4, 5184 ( 4F + 2 ) F (A c 0 ) 19 (A c 0) 2 = 5 72 (A c 0) 2, In such a case 11 the incumbent has no incentive to act covertly, but he always deters entry by means of an overt behavior. Hence, we can say that, under medium-low corruption, Λ 5 72 (A c 0) 2, large levels of the entry fee support limit pricing against the covert alternative of graft payments. Stackelberg versus bribe: As we previously saw, the Stackelberg game is supported by: { 1 F max 9 (A 2c 0) 2, (3 2 } 2) (A c 0 ) 2 and c A This combines two different situations (see Section 4): a significantly positive, although not very large, entry fee when this is accompanied by large capacity costs, c 0 : 1 9 (A 2c 0) 2 F < (3 2 2) (A c 0 ) 2 and A c A low levels of the entry fee, F < 1 9 (A 2c 0) 2, when, in addition, the (relative) capacity cost is medium-large, 1 5 A < c 0 < 1 2 A. When playing Stackelberg, the incumbent gets a level of profit equal to: Π S i = 1 8 (A c 0) 2 Comparing this with the expected profit from playing covertly, we find out that for the incumbent is optimal to act dishonestly whenever: 5 72 (A c 0) 2 Λ (A c 0) 2 When, instead, Λ > (A c 0) 2, the incumbent is better off by acting strategically but overtly. 11 The range of moral costs which support bribery is not comprised in our reference domain. 10

13 In words, this means that the incumbent has incentives to behave covertly when the degree of corruption is significantly high; this is expressed in our model by medium-low moral costs of a covert behavior (low Λ). Note that the range of values Λ that support covert behavior is increasing with the amount (A c 0 ) 2. Hence, the range of values Λ for which offering a bribe to the public official is the incumbent s optimal action is increasing with the value of A and decreasing with the level of the capacity-installing costs, c 0. This happens because larger (market dimensions) A, and/or lower capacity-installing costs increase the monopoly profits more than the leader s Stackelberg ones. This makes more desirable a situation of monopoly and hence, makes it worth taking more risk and behave dishonestly in order to block entry, even under lower levels of corruption (higher Λ 12 ). For lower (A c 0 ) 2 instead, if the employee s moral costs are not sensibly low, the incumbent finds it optimal to act overtly by playing Stackelberg. That is, if lower profits from monopoly are not compensated, by a higher probability of corruption (higher probability of success in deterring entry), then profits from playing Stackelberg are higher than those that the incumbent expects from playing bribe. Analyzing entry, we can say that for large Λ entry occurs for any value of the entry fee considered. This happens because, when the incumbent perceives low degree of corruption in the public sector, he chooses to strategically play Stackelberg instead of offering grafts. On the other hand, for medium-low Λ, that is, when the perceived level of corruption is rather high, the incumbent expects to get more profit by bribing the public employee in exchange for maintaining a monopoly position. In this situation entry happens with a probability P r(reject bribe) = P r(λ > B ) = Λ (A c 0) 2 which is increasing with Λ and decreasing with the difference (A c 0 ). Going further with our analysis, we can say that the negative effects of high corruption are double-sided. The perception of a high level of corruption (low Λ) induces the incumbent to opt for the covert behavior. Then, on one hand, if the moral cost of the public employee, λ, is indeed low, the bribe is accepted and entry does not occur. On the other hand, ifthe public official has a high moral cost, λ > B, he rejects the bribe but still this leads to Cournot competition while an overt behavior would have led to Stackelberg competition which is better off from both the social and consumers perspective. Hence, we can assert that the perception of strong corruption is bad even when it does not reduce competition since it alters the behavior of the incumbent with negative effects on the consumers and society in general. Cournot versus bribe: First of all, recall that when the incumbent acts overtly, it is optimal for him to play a Cournot game only when both the entry fee and the capacity costs are low, F < 1 9 (A 2c 0) 2 and c A 12 This is equivalent to a higher probability that the public employee rejects the bribe. 11

14 In such a situation the optimal overt action for the incumbent is to previously install capacity k = A+c0 3 and to produce a quantity equal to this capacity. The profits that he gets are: Π C 1k = 1 9 (A2 Ac 0 2c 0 ) As above, we are interested if under such levels of the entry fee and capacity cost the incumbent would be better off by fighting overtly or covertly. Comparing the profits that the firm expects in each of the two cases, we obtain that the incumbent is better off by playing Cournot when: Λ 25(A c 0) 4 576c 0 (A 3c 0 ) and finds it optimal to play covertly instead, when Λ is lower than the above threshold which we denote Λ C. In light of the previous results, this one is intuitive: the incumbent finds attractive to play Cournot only when the level of corruption is very low (that is, for very large Λ). Otherwise, for significant probability of corruption, Λ = Λ C ɛ, ɛ > 0, the incumbent expects larger profits from acting dishonestly and hence prefers playing bribe. Similarly to the previous case, we observe that the threshold is increasing with A and decreasing with the values of c 0. This means that the larger is A (the dimension of the market) and/or the lower is the capacity-cost, c 0, the larger is the range of values Λ which induce the incumbent to behave dishonestly. Summing up our findings, we can state the following conclusions: Proposition 2. When the level of corruption is high, bribery reduces competition on the market, substituting limit pricing. When the degree of corruption is low, it has no effect on the market competition. The range of values Λ which support covert behavior is increasing with the dimension of the market, A and decreasing with the capacity installing cost, c 0. Proposition 3. The incentive to choose a covert behavior is stronger when the optimal overt action is Cournot than when the best overt action is Stackelberg which, in turn is stronger than in the case of limit pricing optimality: Λ l > Λ S > Λ C. 5.2 High corruption, Λ < 5 72 (A c 0) 2 In such a case the optimal bribe: B = min {Λ, 572 } (A c 0) 2 = Λ that is, the optimal bribe is equal to the maximum possible moral cost of the public official. The main feature of this situation is that the public employee 12

15 always accepts the bribe proposed to him and, consequently, if the incumbent decides to act covertly, entry is always blocked and the game has a monopoly outcome. This is a rather trivial and very rare situation of a totally corrupt public sector. However, it may still be representative for countries like Sub- Saharan countries where the level of corruption is extremely high. Analyzing quickly this situation, we can immediately say two things: on one hand, for those levels of the entry fee that promote Stackelberg or Cournot as the optimal incumbent s overt choice, the dishonest behavior is always preferred by the incumbent who is certain that paying a bribe of Λ guarantees him a position of monopoly on the market. Certainly, in such situations, the consumers surplus and the social welfare are the monopoly ones and hence lower than in a situation in which the incumbent plays honestly. On the other hand, when the entry fee is very large 13, this leads again to a situation of blockaded monopoly in which the incumbent maintains his position without having to play covertly but he just needs to install production capacity k = q m and produce the monopoly quantity. The most interesting situation here is when the entry fee is significantly high, so that it promotes limit pricing as the optimal strategy in an overt subgame 14. Comparing the profit from a limit pricing action: Π l = 2 F (A c 0 ) 4F to those expected when playing bribery: EΠ B = 1 4 (A c 0) 2 Λ we obtain that the incumbent is better off by playing covertly whenever the probability of corruption is lower than the threshold, Λ: Λ < Λ = 1 4 (A c 0) 2 2 F (A c 0 ) + 4F In words, for those values of F that promote limit quantity as the optimal overt action and for significantly high levels of the parameter Λ, the incumbent finds it less costly to act honestly. When Λ is large, the two combined effects: i) overt entry deterrence is not too costly due to the fact that F is significantly positive (inducing a lower q l ) and ii) the high costs of playing covertly (the bribe is equal to Λ in this case), induce the incumbent to act honestly, on his own. On the contrary, when Λ < Λ, the low cost of playing limit pricing is offset by the even lower cost of deterring entry by paying a bribe. We observe that the threshold Λ is positively related to the difference (A c 0 ), hence is decreasing with the capacity-installing cost, c 0 while it is increasing with the parameter A. This means that the range of moral costs from corruption, Λ, which support a covert behavior of the incumbent is larger when the difference (A c 0 ) is larger. This reflects the fact that larger markets and/or lower 13 See Blockaded monopoly on page See Limit pricing on page 7. 13

16 production costs makes worth paying more in order to maintain an unaltered monopoly position. Summing up, we can formulate the following result: Proposition 4. In an extremely corrupt society, competition is always negated because entry never occurs. The explanation of this result is that in economies with high degree of corruption either the incumbent deters the rival s entry by means of practicing limit pricing or entry is blocked by a corrupt public official who does not issue a production licence to the potential entrant. 6 Social implications We analyze now the consumers surplus and the social welfare under the overt and covert behavior respectively. We compute the social welfare as the total amount of the producers profits and the consumers surpluses. Note that the entry fee does not affect the level of social welfare since it is just a transfer from the entrant to the Government. Then, the social welfare under overt behavior is given by: W = Π i + Π e + S c where Π i stands for the incumbent s profit, Π e for the entrant s gross profit 15 from production and S c represents the consumers surplus. The amount of the social welfare defined as above, varies according to the type of competition in the market (see appendix for detailed formula). Under covert behavior the expected welfare is: EW B = Pr(deny p.l.) (Π m i + Sc m ) + Pr(issue p.l.) (Π C i + Π C e + SC C ) [ 4 = (A c 0 ) ] 5184Λ (A c 0) 2 with low degree of corruption and: EW B h = W m = 3 8 (A c 0) 2 under high corruption. Note that the bribe, being a transfer, does not enter the welfare function directly, but influences welfare only indirectly, by means of its effects on competition (hence on supply and equilibrium prices). Low corruption. When the level of the entry fee supports limit pricing as the optimal overt action, we observe that consumers and society are worse off under bribery when the entry fee is relatively low or when a large entry fee is accompanied by a high level of corruption (low Λ). When, instead, a large level 15 The profit before subtracting the entry fee. 14

17 of the entry fee is accompanied by a low degree of corruption, limit pricing leads to lower levels of welfare and consumers surplus than a covert action would do. The explanation is the following. On one hand, under limit pricing both the consumers surplus and the social welfare depend negatively on the level of the entry fee. This is because the lower is the entry fee, the higher is the production that the incumbent needs to commit to, and to produce in order to deter entry. Consequently, a low entry fee implies a high limit quantity, q l, that is, a larger supply and a lower equilibrium price, increasing the consumers surplus and social welfare. On the other hand, when Λ is significantly large, the probability that the public official rejects the bribe and a Cournot game is played is high. Then, a low entry fee leads to higher welfare under limit pricing than under bribery, but large levels of F accompanied by large Λ induce higher levels of expected welfare under bribery, due to the combined effect of two elements: 1) low levels of q l implying high equilibrium prices of an overt behavior, together with 2) the reduced probability that the public employee behaves dishonestly and, consequently, the high probability of entry under a covert incumbent s behavior. When the optimal incumbent s overt action is committing to Stackelberg competition we easily note that social welfare and consumers surplus are always lower under bribery than under Stackelberg. The explanation is rather simple. In the Stackelberg setup the equilibrium supply is always larger and the price is lower than in both monopoly and Cournot competition. This makes consumers always better off under Stackelberg than both under monopoly or Cournot. Given that the expected welfare under bribery is a linear combination of the welfare under monopoly and the welfare under Cournot competition (where the weights are the probability that the public official accepts and the probability that he rejects the bribe respectively), it is immediate that welfare under Stackelberg is higher than the expected welfare under corruption. This confirms that incumbent s overt actions are more desirable for the society than covert behavior is, even when the probability of succeeding in blocking entry is lower (the probability that the graft is rejected is higher). Finally, when Cournot is the incumbent s optimal overt move, we can immediately prove that both consumers and society are worse off under bribery. The explanation is similar to the Stackelberg case: Cournot where the incumbent has infinite capacity is always preferable from a social point of view to both monopoly and common Cournot. High corruption. We only remark that under high corruption the outcome of a covert behavior is monopoly. Consequently, the consumers surplus and the social welfare are always lower under covert than under overt behavior. Proposition 5. i) When the entry fee F is low the consumers and the society are always better off under the incumbent s overt behavior. ii) However, under large F, supporting limit pricing, and low corruption (high Λ) the social welfare and the consumers surplus are both larger under bribery. iii) When the degree 15

18 of corruption is high an overt behavior is always preferred by the consumers and the society. 7 Extensions In this section we briefly summarize some results, obtained when altering two of our initial assumptions: i) we allow the entrant to bribe as well; ii) we assume that the incumbent has a positive production capacity before the game starts (before date 0). 7.1 Competition in Bribes We explore how our previous results change when we allow both rivals, the incumbent and the entrant, to compete by simultaneously offering a bribe to the public official. First of all, we remark that, given our assumption that the entrant satisfies all legal requirements, it makes sense for him to offer a bribe only when the incumbent fights covertly. When accepted, in exchange for the graft, the public official favors the bribe payer. In a covert subgame, the public employee accepts the highest bribe between the two if this is larger than his moral cost from behaving dishonestly and he rejects both grafts if the highest one is lower than λ. We assume that, when the incumbent and the entrant s bribes are equal, the public official accepts the incumbent s offer 16. Denoting with B and b the bribes offered by the incumbent and the entrant respectively, the utility function of the public employee in a covert subgame is: U pe = B if B max{b, λ} b if b > max{b, λ} 0 if λ > max{b, B} In the equilibrium the entrant offers a bribe: b = Π c F that is, his optimal bribe is equal to his net profit. The incumbent s optimal graft is: { { B = max b, min Λ, 1 }} 2 (Πm Π c ) We remark that the incumbent s optimal bribe is always at least as large as the entrant s. Hence, in a covert subgame either the incumbent s graft is accepted or both bribes are rejected. Note that, as one may expect, the incumbent s optimal bribe in this case is always larger than in the situation described in section It is not unnatural to assume this since, in such cases, the incumbent would effectively bribe b + ɛ, with ɛ > 0, ɛ 0. 16

19 Performing a similar analysis with the one in Section 5, we find out that the results are similar to the ones we obtained for the case of a unique bribe-payer (the incumbent) with the following main variations: 1. Based on the values of the entry fee F, and the maximum moral cost Λ, in this case we can identify four different situations with respect to the optimal incumbent s bribe. a) When Λ 5 72 (A c 0) 2 and F 1 24 (A c 0) 2, that is, medium-low corruption and large entry fee 17, the incumbent s optimal bribe is exactly the one in subsection 5.1, namely, B = 5 72 (A c 0) 2 b) Under relatively low degree of corruption, Λ 1 9 (A c 0) 2 F and low F, F < 1 24 (A c 0) 2, the optimal incumbent s bribe is: B = 1 9 (A c 0) 2 F Λ c) For low moral costs, Λ < 5 72 (A c 0) 2 and a large entry fee, F 1 24 (A c 0) 2, it is immediate that: B = Λ d) For low levels of both moral costs and entry fee, Λ < 1 9 (A c 0) 2 F and F < 1 24 (A c 0) 2, the optimal incumbent s bribe is: B = 1 9 (A c 0) 2 F > Λ 2. By comparing the overt and covert behavior in this case, we find out that, overall, the range of values that support covert fight under bribe competition is at most as large as that in the case of a unique bribe-payer. Indeed, while the outcomes of the cases a) and c) are identical to those in Section 5, in case b) the range of values that support bribery is always lower than its correspondent in Section 5. We get that Λ l CB Λl, Λ S CB ΛS and Λ c CB Λc, where the values Λ CB stand for the threshold that separates overt from covert behavior when both rivals can bribe (CB=competition in bribes). The explanation of this result is the following: when both rivals can bribe, it becomes more costly for the incumbent to fight through bribery since, in the situations described by items b) and d) above, the incumbent has to bribe more than his optimal. Hence, in those situations, covert fight becomes the optimal action only if this supplementary cost is compensated by a lower risk of bribe rejection, that is, by a larger probability of the bribe being accepted (lower values of Λ). 17 For relatively large entry fee, namely F 1 24 (A c 0) 2, we have that 1 9 (A c 0) 2 F 5 72 (A c 0) 2 17

20 7.2 Positive Initial Production Capacity The second extension of our model regards the case in which the incumbent has a positive production capacity before the game starts. Then, at date 0 his problem is to choose between bribing (covert fight) or installing additional production capacity in order to overtly maximize his profits. We start from the natural assumption that an incumbent already operating in the market installed, at the beginning of his activity, the monopoly capacity, that is, the initial capacity is k 0 = 1 2 (A c 0) 2. Under such an assumption, we find that covert fight is played in most situations. The only cases when overt behavior is preferred to the covert one are those in which the entry fee is large, supporting blockaded monopoly or limit pricing and, in addition, corruption is low (Λ is large). Such results are rather intuitive. An incumbent who has already an installed production capacity equal to the monopoly output is extremely powerful on the market. Then, when strategic accomodation is his optimal overt behavior 18, he always prefers the covert fight. The explanation is that, in such cases, i) if the bribe is rejected, the incumbent s initial production capacity still allows him to play the optimal overt action at no extra cost; ii) if the public official accepts the bribe, then the incumbent maintains a monopoly position and he gets, net of the bribe, more than the overt profits. The only case here, in which the incumbent really faces a decision problem between overt and covert fight is when limit pricing is the optimal overt action. This is due to the fact that in such a situation the limit quantity is larger than the installed monopoly capacity, q l > q m. Then, limit pricing beats covert fight whenever the entry fee is relatively large (determining a low q l ) and corruption is low, { 1 max 9 (A 2c 0) 2, (3 2 } 2) (A c 0 ) 2 32 F < 1 16 (A c 0) 2, c 0 > 1 5 A and Λ (A c 0 ) 4 32(A c 0 ) F (A c 0 ) 1024F. 7.3 Unbounded support of the moral costs Finally, we relax the assumption that the moral cost, λ U[0, Λ]. We assume instead, that the support of λ is infinite and that the moral cost of a dishonest behavior which characterizes the public employee follows an exponential distribution, F (x) = Pr(λ x) = 1 e γx where the parameter γ represents the perceived degree of corruption in the economy; it takes positive values, with γ = 0 representing a totally honest society. 18 Stackelberg is optimal when q S (= q m ) < min{q c, q l } and Cournot is the best overt action when q c < min{q S (= q m ), q l } 18

21 Repeating the reasoning in section 3 we find the profit that the incumbent expects from a covert behavior, EΠ B = (1 e γb )(Π m B) + e γb Π c The optimal bribe, B, which maximizes this expected payoff, has to verify the first order condition: γe γb (Π m Π c ) γe γb B 1 + e γb = 0 The following result is proved in subsection 5 of the appendix: Proposition 6. There exists a bribe B (0, (Π m Π c )) which maximizes the expected profits from covert fight. The optimal bribe under infinite support is increasing with (A c 0 ) and decreasing with the level of corruption. This result implies that the optimal bribe is increasing with the dimension of the market and decreases with the capacity-costs. This replicates what we found for the more restrictive case in which λ has a finite support. In addition, the statement above establishes that the larger is the level of corruption in the economy the lower is the optimal bribe that an incumbent should offer in a covert fight. The explanation stands in the fact that the larger is γ the higher is the probability that the public employee has a low moral cost and this increases the probability of the (a lower) bribe being accepted. Going further with our analysis, we compare the expected profit from covert fight with the profit that the incumbent can get from an optimal overt behavior. We find that for every optimal overt action there exists a level of corruption γ o such that EΠ B (γ o ) = Π o, (where the upper index o stands for the optimal overt action of the incumbent). For every level of corruption larger than the threshold, γ > γ o, the incumbent s optimal strategy is covert fight. If, instead, the degree of corruption in the economy is lower than γ o the incumbent prefers overt behavior. This result is exactly what Proposition 2 in Section 5 states. It is not difficult to prove that also the previous Proposition 3 remains valid in this case. We remind that Proposition 3 states that the incentive to fight covertly is stronger, the stronger is the competitive pressure on the market. Mathematically such a result is represented by an ordering of the thresholds. The lowest threshold is that corresponding to an optimal Cournot, followed by the threshold corresponding to Stackelberg and finally by that of limit pricing 19 : γ l > γ S > γ ck The unique result of the previous case which does not hold anymore is Proposition 4 ; in this situation the probability that entry is deterred by corrupt 19 Note that a lower threshold means in this case a larger range of values γ which support covert fight 19

22 behavior is never 1. This is not surprising under the assumption of infinite support for λ, given that it is not possible anymore for the incumbent to bribe the maximum moral cost of the public official, which is infinite. Given the above findings we can conclude that the qualitative results that we obtained under a uniform distribution of λ hold also for the case with infinite support and hence that the previous results are robust to changes in the support and the distribution of the moral cost. 8 Conclusions In the present paper we identify in which situations and to what extent the degree of corruption affects the level of competition on the market. When the entry fee is high the incumbent prefers an overt fight even for high levels of corruption. If the entry fee is not so large as to support blockaded monopoly, covert behavior may become the optimal strategy for an incumbent, when the degree of corruption in the public sector is significantly large. In such cases bribery may reduce competition and hence, it represents a real alternative to limit pricing or a substitute of limit pricing when the optimal overt action is Stackelberg or Cournot. Given a covert behavior of the incumbent, the probability that the entrant is denied the production licence and entry does not occur is: { } 5 P r(no entry) = P r(λ < B ) = min 72Λ (A c 0) 2, 1 decreasing with Λ 20 and increasing with the difference (A c 0 ). For low probability of corruption and low levels of the entry fee (the most desirable case) strategic accomodation is the optimal choice of the incumbent who decides, based on the level of the capacity cost, to play Stackelberg or Cournot. In such situations the alternative covert behavior has no effect on competition. The level of the entry fee can be an extremely important political instrument. Depending on the environment, and we think here mainly about the levels of corruption and the market characteristics: dimensions and production costs, the entry fee can exert two important effects. 1. In economies where the level of corruption is low, practicing low levels of entry fees can be a good instrument to diminish the negative effects of limit pricing. This happens essentially because a low entry fee induces a large limit quantity, q l. The implications are the following: low entry fee may make unattractive a limit pricing strategy for the incumbent; large q l can prove not to be optimal for the incumbent but, instead, to be dominated by a policy of strategic entry accomodation. On the contrary, a high entry fee may lead to situations of blockaded 20 The expected moral cost of a public employee from a dishonest behavior is positively related to Λ. 20